cos(x) 是否需要在使用 IEEE-754 的不同 C++ 实现中返回相同的值？

Question

Is there any sort of guarantee - either in the C++ standard or in some other document - that C++ code computing cos(x) will produce identical values when compiled with g++, clang, MSVC, etc., assuming those implementations are using IEEE-754 64-bit double s and the input value x is exactly equal? My assumption is "yes," but I'd like to confirm that before relying on this behavior.

Context: I'm teaching a course in which students may need to compute trigonometric functions of inputs. I can assure that those inputs are identical when fed into the functions. I'm aware that equality-testing double s is not a good idea, but in this specific case I was wondering if it was safe to do so.

Answer 1

cos is a transcendental function. Transcendental functions are subject to the table-maker's dilemma . Informally, what this means is: let's say you come up with some iterative algorithm for approximating the cosine of an input value: for example, a Taylor series. When you run this iterative algorithm, you have to decide how much extra precision to keep at the intermediate stages (rounding too early may reduce the accuracy of the final result). But because the function is transcendental, it's very difficult to determine how many extra bits must be carried during the calculation in order to yield a correctly rounded final result, and for some input values, the number of extra bits required might be very large.

For this reason, it is generally not practical to design hardware that guarantees correctly rounded results for transcendental functions such as cos (where "correctly rounded" means that the resulting floating point value is the one that's closest to the true real value of the function). Instead, the hardware designers will implement a calculation technique that performs reasonably well and that, for most practical input values, will yield a result that is within 1 bit of the exact real result. (If you absolutely need a cosine function that always yields a correctly rounded result, then apparently it's possible to implement one: GNU MPFR claims to have done it. But this will perform much worse than hardware.)

IEEE 754 (2008) lists cos as one of the "recommended correctly rounded functions", which means that if you implement IEEE 754's version of cos , then you have to yield a correctly rounded result. But these functions are only "recommended" to be provided, and not required. Therefore, a conforming implementation of IEEE 754 might not provide a correctly rounded cos function, and might instead provide a "practical" cos function as described in the previous paragraph. Therefore, in practice, two implementations of C++ which both claim to be IEEE 754 compliant may not yield the exact same value for a transcendental function such as cos when applied to the same argument.

(Note that IEEE 754 requires implementations to provide a square root function that is correctly rounded. This is not a transcendental function, so correctly rounding it is not nearly as difficult.)